The state of the art is summarized in the document of PCT 200603237, which is the only one known document that aims at solving the problem in such a way for two channels using two pairs of orthogonal complementary sequences.
Communication systems, spectrum analysis, RADAR, SONAR and other systems of characterization and identification transmit a signal that arrives—reflected or not—to the receiver after passing through a transmission means. This means acts as a linear filter with an impulse response in frequencies H (ω) or temporary h[n]. In order to make the process of retrieving emitted information possible, in most communication systems it is essential to eliminating effects produced by the transmission means in the emitted signal s[n]. This process is known as equalization. The response in frequency can also be used to make a special analysis of the means and, thus, obtaining information of its physical properties.
The channel acts as a filter and distorts the signal. Noise, n[n], due to channel disruptions, thermal noise and other signals interfering with those emitted should be added as well. In conclusion, the signal received, r[n], can be modeled as follows:r[n]=s[n]*h[h]+n[n]  (1)Where * denotes the convolution operation.
By generalizing equation (1) for the case of P inputs and Q outputs, rejecting the term of noise n[n] by clarity, signals r[n]and s[n] are vectors of Q and P length respectively, and h[n] corresponds to a matrix of P×Q size, whose elements correspond to vectors of the functions of cross transference among each input and each output of the system.
Thus, the previous expression remains in the time domain as:r [n]=s[n]*hP,Q[n]  (2)Where * is the convolution operator, 0 in the frequency domain as the productR=S·HP,Q  (3)
Where channel's transference matrices are:
                              H                      P            ,            Q                          =                                            (                                                                                          H                                              1                        ,                        1                                                                                                  …                                                                              H                                              1                        ,                        Q                                                                                                                                  …                                                                              H                                              p                        ,                        q                                                                                                  …                                                                                                              H                                              P                        ,                        1                                                                                                  …                                                                              H                                              P                        ,                        Q                                                                                                        )                        ⁢                                                  ⁢                          h                              P                ,                Q                                              =                      (                                                                                H                                          1                      ,                      1                                                                                        K                                                                      H                                          1                      ,                      Q                                                                                                                    M                                                  O                                                  M                                                                                                  H                                          P                      ,                      1                                                                                        L                                                                      H                                          P                      ,                      Q                                                                                            )                                              (        4        )            
For that, numerous identification methods perform a sequence emission with each of the transmitters separately to identify each coefficient of said transference matrix and thus avoiding mutual interference. Hence, identification time increases with the value of P, Q and the length of the channel to be identified.
The objective of the improvement proposed is to obtain all values of channel coefficients HP,Q or hP,Q[N] as fast as possible using the technology employed in the previous patent for two channels but extending it to multiple simultaneous channels.
Like the previous patent, the fundamental base is the use of complementary sequences properties, whose elements or sequences belonging to orthogonal families meet the following properties:
                                                        s                              i                ,                j                                      ⊗                          s                                                i                  ′                                ,                                  j                  ′                                                              =                                                    ϕ                                                      s                                          i                      ,                      j                                                        ⁢                                      s                                                                  i                        ′                                            ,                                              j                        ′                                                                                                        ⁢                                                          [              k              ]                        =                                          1                L                            ⁢                                                ∑                                      n                    =                    1                                    L                                ⁢                                                                            s                      ij                                        ⁡                                          [                      n                      ]                                                        ·                                                            s                                                                        i                          ′                                                ,                                                  j                          ′                                                                                      ⁡                                          [                                              n                        +                        k                                            ]                                                                                                          ⁢                                  ⁢                              S            ⁢                                                  ⁢            A            ⁢                                                  ⁢            C            ⁢                                                  ⁢                          F              ⁡                              [                k                ]                                              =                                                    ∑                                  j                  =                  1                                M                            ⁢                                                ϕ                                                            s                                              i                        ,                        j                                                              ⁢                                          s                                              i                        ,                        j                                                                                            ⁢                                                                  [                k                ]                                      =                          M              ·                              δ                ⁡                                  [                  k                  ]                                                                    ⁢                                  ⁢                              S            ⁢                                                  ⁢            C            ⁢                                                  ⁢            C            ⁢                                                  ⁢                          F              ⁡                              [                k                ]                                              =                                                    ∑                                  j                  =                  1                                M                            ⁢                                                ϕ                                                            s                                              i                        ,                        j                                                              ⁢                                          s                                                                        i                          ′                                                ,                        j                                                                                            ⁢                                                                  [                k                ]                                      =                          0              ⁢                                                          ⁢                              ∀                k                                                                        (        5        )            
Where {circle around (x)} is the correlation operator, SACF is the sum of autocorrelation functions of set Si and SCCF is the sum of cross-correlations of two sets Si and Si,; both sets are uncorrelated and formed by sequences Si,j corresponding to the sequence j-th of set Si.